Here, thousands of small farmers straddle the line that divides the six counties of Ulster from the Republic of Ireland, but when Brexit is complete that land will be instantly split between two different regulatory and trading regimes: on the same farm EU cows will graze in one field, and British ones in another.

What that will mean for the Irish dairy industry is the question troubling Gabriel D’Arcy, the chief executive of LacPatrick, a milk-co-operative...